Based on the available literature, we considered the plausible existence of five or more bases in the earliest DNA molecules. Our algebraic and biological model suggests the plausibility of the transition from a primeval genetic code with an extended DNA alphabet to the present standard genetic code, where the symbol represents one or more hypothetical bases with unspecific pairings. The results suggest that the Watson–Crick base pairing ( and ) and the non-specific base pairing of the hypothetical ancestral base used to define the sum and product operations are enough features to determine the coding constraints of the primeval and the modern genetic code, as well as the transition from the former to the latter. So then, it is algebraically proved that the present genetic code architecture (very well described in the middle of the last century by Crick and other authors) could be derived from the plausible ancient architecture as quotients of three-dimensional vector spaces and as quotient groups. The coordinate representation of ancient and present codons allows us to insert the set into the three-dimensional real vector space that can be represented as an ordinary cube (or regular hexahedron) with three of its faces contained in the coordinate planes , , . This Demonstration lets you manipulate the algebraic operations over the DNA base triplets. You can visualize the principal algebraic features of the ancient (plausible) and standard genetic codes [1]. In particular, the algebraic operations over the codon subsets formed by vertical planes, lines, and amino acids (or individual codons) can be visualized. The cube here can be seen as containing biological and algebraic criteria.